Is it obligatory to repeat prayers for more than a month that were performed facing the wrong direction without ijtihad, and can they be made up all at once?

The worshipper must face the Qibla. If he does not know its direction, he must exert effort to determine it. If someone prays while doubting the Qibla's direction, and is capable of exerting effort, then his prayer is not valid, and he must repeat it. However, if someone exerted effort and made a mistake in the Qibla's direction, his prayer is valid, and he does not need to repeat it. There are many ways to exert effort to determine the Qibla, including using the compass available on phones. If you were negligent in using the compass, despite knowing its importance, or did not notice its presence and relied on comparing your new location to an old one, then we hope there is no harm upon you. If you fear negligence in your investigation, then it is good to be cautious by repeating the missed prayers in order and immediately.

In summary: Whoever exerted a permissible effort and then it became clear that he erred in facing the Qibla, his prayer is valid. But whoever was negligent in exerting effort must repeat the prayers in order. And if he feared falling short and made up what he missed, that is good.